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Mathematical Background:Foundations of Infinitesimal Calculus

second edition

by

K. D. Stroyan

x

y

y=f(x)

dx

dy

x

dx

dy

Figure 0.1: A Microscopic View of the Tangent

Copyright c1997 by Academic Press, Inc. - All rights reserved.

Typeset with AMS-TEX


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i

Preface to the Mathematical Background

We want you to reason with mathematics. We are not trying to get everyone to giveformalized proofs in the sense of contemporary mathematics; proof in this course meansconvincing argument. We expect you to use correct reasoning and to give careful expla-nations. The projects bring out these issues in the way we find best for most students,but the pure mathematical questions also interest some students. This book of mathemat-ical background shows how to fill in the mathematical details of the main topics fromthe course. These proofs are completely rigorous in the sense of modern mathematics technically bulletproof. We wrote this book of foundations in part to provide a convenientreference for a student who might like to see the theorem - proof approach to calculus.

We also wrote it for the interested instructor. In re-thinking the presentation of beginningcalculus, we found that a simpler basis for the theory was both possible and desirable. Thepointwise approach most books give to the theory of derivatives spoils the subject. Clearsimple arguments like the proof of the Fundamental Theorem at the start of Chapter 5 below

are not possible in that approach. The result of the pointwise approach is that instructorsfeel they have to either be dishonest with students or disclaim good intuitive approximations.This is sad because it makes a clear subject seem obscure. It is also unnecessary by andlarge, the intuitive ideas work provided your notion of derivative is strong enough. Thisbook shows how to bridge the gap between intuition and technical rigor.

A function with a positive derivative ought to be increasing. After all, the slope ispositive and the graph is supposed to look like an increasing straight line. How could thefunction NOT be increasing? Pointwise derivatives make this bizarre thing possible - apositive derivative of a non-increasing function. Our conclusion is simple. That definitionis WRONG in the sense that it does NOT support the intended idea.

You might agree that the counterintuitive consequences of pointwise derivatives are un-fortunate, but are concerned that the traditional approach is more general. Part of thepoint of this book is to show students and instructors that nothing of interest is lost and a

great deal is gained in the straightforward nature of the proofs based on uniform deriva-tives. It actually is not possible to give a formula that is pointwise differentiable and notuniformly differentiable. The pieced together pointwise counterexamples seem contrivedand out-of-place in a course where students are learning valuable new rules. It is a theoremthat derivatives computed by rules are automatically continuous where defined. We wantthe course development to emphasize good intuition and positive results. This backgroundshows that the approach is sound.

This book also shows how the pathologies arise in the traditional approach we leftpointwise pathology out of the main text, but present it here for the curious and for com-parison. Perhaps only math majors ever need to know about these sorts of examples, butthey are fun in a negative sort of way.

This book also has several theoretical topics that are hard to find in the literature. Itincludes a complete self-contained treatment of Robinsons modern theory of infinitesimals,

first discovered in 1961. Our simple treatment is due to H. Jerome Keisler from the 1970s.Keislers elementary calculus using infinitesimals is sadly out of print. It used pointwisederivatives, but had many novel ideas, including the first modern use of a microscope todescribe the derivative. (The lHospital/Bernoulli calculus text of 1696 said curves consistof infinitesimal straight segments, but I do not know if that was associated with a magni-fying transformation.) Infinitesimals give us a very simple way to understand the uniform


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ii

derivatives, although this can also be clearly understood using function limits as in the textby Lax, et al, from the 1970s. Modern graphical computing can also help us see graphsconverge as stressed in our main materials and in the interesting Uhl, Porta, Davis, Calculus& Mathematica text.

Almost all the theorems in this book are well-known old results of a carefully studiedsubject. The well-known ones are more important than the few novel aspects of the book.However, some details like the converse of Taylors theorem both continuous and discrete are not so easy to find in traditional calculus sources. The microscope theorem for differentialequations does not appear in the literature as far as we know, though it is similar to researchwork of Francine and Marc Diener from the 1980s.

We conclude the book with convergence results for Fourier series. While there is nothingnovel in our approach, these results have been lost from contemporary calculus and deserveto be part of it. Our development follows Courants calculus of the 1930s giving wonderfulresults of Dirichlets era in the 1830s that clearly settle some of the convergence mysteriesof Euler from the 1730s. This theory and our development throughout is usually easy to

apply. Clean theory should be the servant of intuition building on it and making itstronger and clearer.There is more that is novel about this book. It is free and it is not a book since it is

not printed. Thanks to small marginal cost, our publisher agreed to include this electronictext on CD at no extra cost. We also plan to distribute it over the world wide web. Wehope our fresh look at the foundations of calculus will stimulate your interest. Decide foryourself whats the best way to understand this wonderful subject. Give your own proofs.


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Contents

Part 1Numbers and Functions

Chapter 1. Numbers 3

1.1 Field Axioms 3

1.2 Order Axioms 6

1.3 The Completeness Axiom 7

1.4 Small, Medium and Large Numbers 9

Chapter 2. Functional Identities 17

2.1 Specific Functional Identities 17

2.2 General Functional Identities 18

2.3 The Function Extension Axiom 21

2.4 Additive Functions 24

2.5 The Motion of a Pendulum 26

Part 2Limits

Chapter 3. The Theory of Limits 31

3.1 Plain Limits 32

3.2 Function Limits 34

3.3 Computation of Limits 37

Chapter 4. Continuous Functions 43

4.1 Uniform Continuity 43

4.2 The Extreme Value Theorem 44

iii


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iv Contents

4.3 Bolzanos Intermediate Value Theorem 46

Part 3

1 Variable Differentiation

Chapter 5. The Theory of Derivatives 49

5.1 The Fundamental Theorem: Part 1 49

5.1.1 Rigorous Infinitesimal Justification 525.1.2 Rigorous Limit Justification 53

5.2 Derivatives, Epsilons and Deltas 53

5.3 Smoothness Continuity of Function and Derivative 545.4 Rules Smoothness 565.5 The Increment and Increasing 57

5.6 Inverse Functions and Derivatives 58

Chapter 6. Pointwise Derivatives 69

6.1 Pointwise Limits 69

6.2 Pointwise Derivatives 72

6.3 Pointwise Derivatives Arent Enough for Inverses 76

Chapter 7. The Mean Value Theorem 79

7.1 The Mean Value Theorem 79

7.2 Darbouxs Theorem 83

7.3 Continuous Pointwise Derivatives are Uniform 85

Chapter 8. Higher Order Derivatives 87

8.1 Taylors Formula and Bending 87

8.2 Symmetric Differences and Taylors Formula 89

8.3 Approximation of Second Derivatives 91

8.4 The General Taylor Small Oh Formula 92

8.4.1 The Converse of Taylors Theorem 95

8.5 Direct Interpretation of Higher Order Derivatives 98

8.5.1 Basic Theory of Interpolation 998.5.2 Interpolation where f is Smooth 101

8.5.3 Smoothness From Differences 102Part 4

Integration

Chapter 9. Basic Theory of the Definite Integral 109

9.1 Existence of the Integral 110


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Contents v

9.2 You Cant Always Integrate Discontinuous Functions 114

9.3 Fundamental Theorem: Part 2 116

9.4 Improper Integrals 119

9.4.1 Comparison of Improper Integrals 1219.4.2 A Finite Funnel with Infinite Area? 123

Part 5Multivariable Differentiation

Chapter 10. Derivatives of Multivariable Functions 127

Part 6Differential Equations

Chapter 11. Theory of Initial Value Problems 131

11.1 Existence and Uniqueness of Solutions 131

11.2 Local Linearization of Dynamical Systems 135

11.3 Attraction and Repulsion 141

11.4 Stable Limit Cycles 143

Part 7Infinite Series

Chapter 12. The Theory of Power Series 147

12.1 Uniformly Convergent Series 149

12.2 Robinsons Sequential Lemma 151

12.3 Integration of Series 152

12.4 Radius of Convergence 154

12.5 Calculus of Power Series 156

Chapter 13. The Theory of Fourier Series 159

13.1 Computation of Fourier Series 160

13.2 Convergence for Piecewise Smooth Functions 167

13.3 Uniform Convergence for Continuous Piecewise Smooth Functions 173

13.4 Integration of Fourier Series 175


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-4

-2

2

4

x

Part 1

Numbers and Functions


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CHAPTER

1 NumbersThis chapter gives the algebraic laws of the number systems usedin calculus.

Numbers represent various idealized measurements. Positive integers may count items,fractions may represent a part of an item or a distance that is part of a fixed unit. Distancemeasurements go beyond rational numbers as soon as we consider the hypotenuse of a righttriangle or the circumference of a circle. This extension is already in the realm of imaginedperfect measurements because it corresponds to a perfectly straight-sided triangle withperfect right angle, or a perfectly round circle. Actual real measurements are always rationaland have some error or uncertainty.

The various imaginary aspects of numbers are very useful fictions. The rules of com-putation with perfect numbers are much simpler than with the error-containing real mea-surements. This simplicity makes fundamental ideas clearer.

Hyperreal numbers have teeny tiny numbers that will simplify approximation estimates.Direct computations with the ideal numbers produce symbolic approximations equivalent

to the function limits needed in differentiation theory (that the rules of Theorem 1.12 givea direct way to compute.) Limit theory does not give the answer, but only a way to justifyit once you have found it.

1.1 Field AxiomsThe laws of algebra follow from the field axioms. This means that algebrais the same with Dedekinds real numbers, the complex numbers, andRobinsons hyperreal numbers.

3


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4 1. Numbers

Axiom 1.1. Field AxiomsA field of numbers is any set of objects together with two operations, addition

and multiplication where the operations satisfy: The commutative laws of addition and multiplication,a1 + a2 = a2 + a1 & a1 a2 = a2 a1

The associative laws of addition and multiplication,a1 + (a2 + a3) = (a1 + a2) + a3 & a1 (a2 a3) = (a1 a2) a3

The distributive law of multiplication over addition,a1 (a2 + a3) = a1 a2 + a1 a3

There is an additive identity, 0, with 0 + a = a for every number a. There is an multiplicative identity, 1, with 1 a = a for every number a = 0. Each number a has an additive inverse, a, with a + (a) = 0. Each nonzero number a has a multiplicative inverse,

1a , with a

1a = 1.

A computation needed in calculus is

Example 1.1. The Cube of a Binomial

(x + x)3 = x3 + 3x2x + 3xx2 + x3

= x3 + 3x2x + (x(3x + x)) x

We analyze the term = (x(3x + x)) in differentiation.

The reader could laboriously demonstrate that only the field axioms are needed to performthe computation. This means it holds for rational, real, complex, or hyperreal numbers.

Here is a start. Associativity is needed so that the cube is well defined, or does not dependon the order we multiply. We use this in the next computation, then use the distributiveproperty, the commutativity and the distributive property again, and so on.

(x + x)3 = (x + x)(x + x)(x + x)

= (x + x)((x + x)(x + x))

= (x + x)((x + x)x + (x + x)x)

= (x + x)((x2 + xx) + (xx + x2))

= (x + x)(x2 + xx + xx + x2)

= (x + x)(x2 + 2xx + x2)

= (x + x)x2 + (x + x)2xx + (x + x)x2)

...

The natural counting numbers 1, 2, 3, . . . have operations of addition and multiplication,but do not satisfy all the properties needed to be a field. Addition and multiplication dosatisfy the commutative, associative, and distributive laws, but there is no additive inverse


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Field Axioms 5

0 in the counting numbers. In ancient times, it was controversial to add this element thatcould stand for counting nothing, but it is a useful fiction in many kinds of computations.

The negative integers

1,

2,

3, . . . are another idealization added to the natural num-bers that make additive inverses possible - they are just new numbers with the neededproperty. Negative integers have perfectly concrete interpretations such as measurementsto the left, rather than the right, or amounts owed rather than earned.

The set of all integers; positive, negative, and zero, still do not form a field because thereare no multiplicative inverses. Fractions, 1/2, 1/3, . . . are the needed additional inverses.When they are combined with the integers through addition, we have the set of all rationalnumbers of the form p/q for natural numbers p and q = 0. The rational numbers are afield, that is, they satisfy all the axioms above. In ancient times, rationals were sometimesconsidered only operators on actual numbers like 1, 2, 3, . . ..

The point of the previous paragraphs is simply that we often extend one kind of numbersystem in order to have a new system with useful properties. The complex numbers extendthe field axioms above beyond the real numbers by adding a number i that solves the

equation x2

= 1. (See the CD Chapter 29 of the main text.) Hundreds of years ago thisnumber was controversial and is still called imaginary. In fact, all numbers are usefulconstructs of our imagination and some aspects of Dedekinds real numbers are muchmore abstract than i2 = 1. (For example, since the reals are uncountable, most realnumbers have no description what-so-ever.)

The rationals are not complete in the sense that the linear measurement of the sideof an equilateral right triangle (

2) cannot be expressed as p/q for p and q integers. In

Section 1.3 we complete the rationals to form Dedekinds real numbers. These numberscorrespond to perfect measurements along an ideal line with no gaps.

The complex numbers cannot be ordered with a notion of smaller than that is compat-ible with the field operations. Adding an ideal number to serve as the square root of1 isnot compatible with the square of every number being positive. When we make extensionsbeyond the real number system we need to make choices of the kind of extension depending

on the properties we want to preserve.Hyperreal numbers allow us to compute estimates or limits directly, rather than making

inverse proofs with inequalities. Like the complex extension, hyperreal extension of the realsloses a property; in this case completeness. Hyperreal numbers are explained beginning inSection 1.4 below and then are used extensively in this background book to show how manyintuitive estimates lead to simple direct proofs of important ideas in calculus.

The hyperreal numbers (discovered by Abraham Robinson in 1961) are still controver-sial because they contain infinitesimals. However, they are just another extended modernnumber system with a desirable new property. Hyperreal numbers can help you understandlimits of real numbers and many aspects of calculus. Results of calculus could be provedwithout infinitesimals, just as they could be proved without real numbers by using onlyrationals. Many professors still prefer the former, but few prefer the latter. We believe thatis only because Dedekinds real numbers are more familiar than Robinsons, but we will

make it clear how both approaches work as a theoretical background for calculus.There is no controversy concerning the logical soundness of hyperreal numbers. The use

of infinitesimals in the early development of calculus beginning with Leibniz, continuing withEuler, and persisting to the time of Gauss was problematic. The founders knew that theiruse of infinitesimals was logically incomplete and could lead to incorrect results. Hyperrealnumbers are a correct treatment of infinitesimals that took nearly 300 years to discover.


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6 1. Numbers

With hindsight, they also have a simple description. The Function Extension Axiom 2.1explained in detail in Chapter 2 was the missing key.

Exercise set 1.1

1. Show that the identity numbers 0 and 1 are unique. (HINT: Suppose 0 + a = a. Adda to both sides.)

2. Show that 0 a = 0. (HINT: Expand0 + ba a with the distributive law and show that0 a + b = b. Then use the previous exercise.)

3. The inverses a and 1a are unique. (HINT: Suppose not, 0 = a a = a + b. Addato both sides and use the associative property.)

4. Show that1 a = a. (HINT: Use the distributive property on 0 = (1 1) a and usethe uniqueness of the inverse.)

5. Show that (1) (1) = 1.6. Other familiar properties of algebra follow from the axioms, for example, if a3 = 0 and

a4 = 0, thena1 + a2

a3=

a1a3

+a2a3

,a1 a2a3 a4 =

a1a3

a2a4

& a3 a4 = 0

1.2 Order AxiomsEstimation is based on the inequality of the real numbers.

One important representation of rational and real numbers is as measurements of distancealong a line. The additive identity 0 is located as a starting point and the multiplicativeidentity 1 is marked off (usually to the right on a horizontal line). Distances to the rightcorrespond to positive numbers and distances to the left to negative ones. The inequality< indicates which numbers are to the left of others. The abstract properties are as follows.

Axiom 1.2. Ordered Field AxiomsA a number system is an ordered field if it satisfies the field Axioms 1.1 and has arelation < that satisfies:

Every pair of numbers a and b satisfies exactly one of the relationsa = b, a < b, or b < a

If a < b and b < c, then a < c.

If a < b, then a + c < b + c. If 0 < a and 0 < b, then 0 < a b.

These axioms have simple interpretations on the number line. The first order axiom saysthat every two numbers can be compared; either two numbers are equal or one is to the leftof the other.


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The Completeness Axiom 7

The second axiom, called transitivity, says that ifa is left ofb and b is left of c, then a isleft of c.

The third axiom says that if a is left of b and we move both by a distance c, then theresults are still in the same left-right order.

The fourth axiom is the most difficult abstractly. All the compatibility with multiplicationis built from it.

The rational numbers satisfy all these axioms, as do the real and hyperreal numbers. Thecomplex numbers cannot be ordered in a manner compatible with the operations of additionand multiplication.

Definition 1.3. Absolute ValueIf a is a nonzero number in an ordered field, |a| is the larger of a anda, that is,|a| = a ifa < a and |a| = a if a < a. We let |0| = 0.

Exercise set 1.2

1. If 0 < a, show thata < 0 by using the additive property.2. Show that 0 < 1. (HINT: Recall the exercise that (1) (1) = 1 and argue by contra-

diction, supposing 0 < 1.)3. Show that a a > 0 for every a = 0.4. Show that there is no order < on the complex numbers that satisfies the ordered field

axioms.

5. Prove that if a < b and c > 0, then c a < c b.Prove that if 0 < a < b and 0 < c < d, then c a < d b.

1.3 The Completeness AxiomDedekinds real numbers represent points on an ideal line with no gaps.

The number

2 is not rational. Suppose to the contrary that

2 = q/r for integers qand r with no common factors. Then 2r2 = q2. The prime factorization of both sides mustbe the same, but the factorization of the squares have an even number distinct primes oneach side and the 2 factor is left over. This is a contradiction, so there is no rational number

whose square is 2.A length corresponding to

2 can be approximated by (rational) decimals in various

ways, for example, 1 < 1.4 < 1.41 < 1.414 < 1.4142 < 1.41421 < 1.414213 < . . .. Thereis no rational for this sequence to converge to, even though it is trying to converge. Forexample, all the terms of the sequence are below 1.41422 < 1.4143 < 1.415 < 1.42 < 1.5 < 2.Even without remembering a fancy algorithm for finding square root decimals, you can test


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8 1. Numbers

the successive decimal approximations by squaring, for example, 1.414212 = 1.9999899241and 1.414222 = 2.0000182084.

It is perfectly natural to add a new number to the rationals to stand for the limit ofthe better and better approximations to

2. Similarly, we could devise approximations

to and make the number that stands for the limit of such successive approximations.We would like a method to include all such possible limits without having to specify theparticular approximations. Dedekinds approach is to let the real numbers be the collectionof all cuts on the rational line.

Definition 1.4. A Dedekind CutA cut in an ordered field is a pair of nonempty sets A and B so that:

Every number is either in A or B. Every a in A is less than every b in B.

We may think of

2 defining a cut of the rational numbers where A consists of all rationalnumbers a with a < 0 or a2 < 2 and B consists of all rational numbers b with b2 > 2. Thereis a gap in the rationals where we would like to have

2. Dedekinds real numbers fill

all such gaps. In this case, a cut of real numbers would have to have

2 either in A or inB.

Axiom 1.5. Dedekind CompletenessThe real numbers are an ordered field such that if A and B form a cut in thosenumbers, there is a number r such that r is in either A or in B and all other thenumbers in A satisfy a < r and in B satisfy r < b.

In other words, every cut on the real line is made at some specific number r, so thereare no gaps. This seems perfectly reasonable in cases like

2 and where we know specific

ways to describe the associated cuts. The only drawback to Dedekinds number systemis that every cut is not a very concrete notion, but rather relies on an abstract notionof every set. This leads to some paradoxical facts about cuts that do not have specificdescriptions, but these need not concern us. Every specific cut has a real number in themiddle.

Completeness of the reals means that approximation procedures that are improvingconverge to a number. We need to be more specific later, but for example, bounded in-creasing or decreasing sequences converge and Cauchy sequences converge. We will not

describe these details here, but take them up as part of our study of limits below.

Completeness has another important consequence, the Archimedean Property Theo-rem 1.8. We take that up in the next section. The Archimedean Property says precisely thatthe real numbers contain no positive infinitesimals. Hyperreal numbers extend the reals byincluding infinitesimals. (As a consequence the hyperreals are not Dedekind complete.)


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Small, Medium and Large Numbers 9

1.4 Small, Medium and Large Num-

bersHyperreal numbers give us a way to simplify estimation by adding infinites-imal numbers to the real numbers.

We want to have three different intuitive sizes of numbers, very small, medium size, andvery large. Most important, we want to be able to compute with these numbers using thesame rules of algebra as in high school and separate the small parts of our computation.Hyperreal numbers give us these computational estimates. Hyperreal numbers satisfy threeaxioms which we take up separately below, Axiom 1.7, Axiom 1.9, and Axiom 2.1.

As a first intuitive approximation, we could think of these scales of numbers in terms of

the computer screen. In this case, medium numbers might be numbers in the range -999 to+ 999 that name a screen pixel. Numbers closer than one unit could not be distinguished bydifferent screen pixels, so these would be tiny numbers. Moreover, two medium numbersa and b would be indistinguishably close, a b, if their difference was a tiny number lessthan a pixel. Numbers larger in magnitude than 999 are too big for the screen and couldbe considered huge.

The screen distinction sizes of computer numbers is a good analogy, but there are diffi-culties with the algebra of screen - size numbers. We want to have ordinary rules of algebraand the following properties of approximate equality. For now, all you should think of isthat means approximately equals.

(a) If p and q are medium, so are p + q and p q.(b) If and are tiny, so is + , that is, 0 and 0 implies + 0.(c) If

0 and q is medium, then q

0.

(d) 1/0 is still undefined and 1/x is huge only when x 0.You can see that the computer number idea does not quite work, because the approximationrules dont always apply. Ifp = 15.37 and q = 32.4, then pq = 497.998 498, mediumtimes medium is medium, however, if p = 888 and q = 777, then p q is no longer screensize...

The hyperreal numbers extend the real number system to include ideal numbers thatobey these simple approximation rules as well as the ordinary rules of algebra and trigonom-etry. Very small numbers technically are called infinitesimals and what we shall assume thatis different from high school is that there are positive infinitesimals.

Definition 1.6. Infinitesimal NumberA number in an ordered field is called infinitesimal if it satisfies

12

> 13

> 14

> > 1m

> > ||for any ordinary natural counting numberm = 1, 2, 3, . We write a b and saya is infinitely close to b if the number b a 0 is infinitesimal.

This definition is intended to include 0 as infinitesimal.


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10 1. Numbers

Axiom 1.7. The Infinitesimal AxiomThe hyperreal numbers contain the real numbers, but also contain nonzero infinites-

imal numbers, that is, numbers 0, positive, > 0, but smaller than all the realpositive numbers.

This stands in contrast to the following result.

Theorem 1.8. The Archimedean PropertyThe hyperreal numbers are not Dedekind complete and there are no positive in-

finitesimal numbers in the ordinary reals, that is, ifr > 0 is a positive real number,then there is a natural counting number m such that 0 < 1m < r.

Proof:

We define a cut above all the positive infinitesimals. The set A consists of all numbers asatisfying a < 1/m for every natural counting number m. The set B consists of all numbersb such that there is a natural number m with 1/m < b. The pair A, B defines a Dedekind

cut in the rationals, reals, and hyperreal numbers. If there is a positive in A, then therecannot be a number at the gap. In other words, there is no largest positive infinitesimal orsmallest positive non-infinitesimal. This is clear because < + and 2 is still infinitesimal,while if is in B, /2 < must also be in B.

Since the real numbers must have a number at the gap, there cannot be any positiveinfinitesimal reals. Zero is at the gap in the reals and every positive real number is in B.This is what the theorem asserts, so it is proved. Notice that we have also proved that thehyperreals are not Dedekind complete, because the cut in the hyperreals must have a gap.

Two ordinary real numbers, a and b, satisfy a b only if a = b, since the ordinary realnumbers do not contain infinitesimals. Zero is the only real number that is infinitesimal.

If you prefer not to say infinitesimal, just say is a tiny positive number and thinkof as close enough for the computations at hand. The computation rules above are stillimportant intuitively and can be phrased in terms of limits of functions if you wish. Theintuitive rules help you find the limit.

The next axiom about the new hyperreal numbers says that you can continue to dothe algebraic computations you learned in high school.

Axiom 1.9. The Algebra Axiom (Including < rules.)The hyperreal numbers are an ordered field, that is, they obey the same rules of

ordered algebra as the real numbers, Axiom 1.1 and Axiom 1.2.

The algebra of infinitesimals that you need can be learned by working the examples andexercises in this chapter.

Functional equations like the addition formulas for sine and cosine or the laws of logsand exponentials are very important. (The specific high school identities are reviewed inthe main text CD Chapter 28 on High School Review.) The Function Extension Axiom 2.1

shows how to extend the non-algebraic parts of high school math to hyperreal numbers.This axiom is the key to Robinsons rigorous theory of infinitesimals and it took 300 yearsto discover. You will see by working with it that it is a perfectly natural idea, as hindsightoften reveals. We postpone that to practice with the algebra of infinitesimals.

Example 1.2. The Algebra of Small Quantities


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Lets re-calculate the increment of the basic cubic using the new numbers. Since the rulesof algebra are the same, the same basic steps still work (see Example 1.1), except now wemay take x any number and x an infinitesimal.

Small Increment of f[x] = x3

f[x + x] = (x + x)3 = x3 + 3x2x + 3xx2 + x3

f[x + x] = f[x] + 3x2 x + (x[3x + x]) x

f[x + x] = f[x] + f[x] x + x

with f[x] = 3x2 and = (x[3x + x]). The intuitive rules above show that 0 wheneverx is finite. (See Theorem 1.12 and Example 1.8 following it for the precise rules.)

Example 1.3. Finite Non-Real Numbers

The hyperreal numbers obey the same rules of algebra as the familiar numbers from highschool. We know that r+ > r, whenever > 0 is an ordinary positive high school number.

(See the addition property of Axiom 1.2.) Since hyperreals satisfy the same rules of algebra,we also have new finite numbers given by a high school number r plus an infinitesimal,

a = r + > r

The number a = r + is different from r, even though it is infinitely close to r. Since issmall, the difference between a and r is small

0 < a r = 0 or a r but a = rHere is a technical definition of finite or limited hyperreal number.

Definition 1.10. Limited and Unlimited Hyperreal NumbersA hyperreal numberx is said to be finite (or limited) if there is an ordinary natural

number m = 1, 2, 3, so that |x| < m.If a number is not finite, we say it is infinitely large (or unlimited).

Ordinary real numbers are part of the hyperreal numbers and they are finite becausethey are smaller than the next integer after them. Moreover, every finite hyperreal numberis near an ordinary real number (see Theorem 1.11 below), so the previous example is themost general kind of finite hyperreal number there is. The important thing is to learn tocompute with approximate equalities.

Example 1.4. A Magnified View of the Hyperreal Line

Of course, infinitesimals are finite, since 0 implies that || < 1. The finite numbers arenot just the ordinary real numbers and the infinitesimals clustered near zero. The rules of

algebra say that if we add or subtract a nonzero number from another, the result is a differentnumber. For example, < < +, when 0 < 0. These are distinct finite hyperrealnumbers but each of these numbers differ by only an infinitesimal, + . Ifwe plotted the hyperreal number line at unit scale, we could only put one dot for all three.However, if we focus a microscope of power 1/ at we see three points separated by unitdistances.


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12 1. Numbers

X 1/d

Pi Pi + dPi - d

Figure 1.1: Magnification at Pi

The basic fact is that finite numbers only differ from reals by an infinitesimal. (This isequivalent to Dedekinds Completeness Axiom.)

Theorem 1.11. Standard Parts of Finite NumbersEvery finite hyperreal number x differs from some ordinary real number r by aninfinitesimal amount, xr 0 or x r. The ordinary real number infinitely nearx is called the standard part of x, r = st(x).

Proof:

Suppose x is a finite hyperreal. Define a cut in the real numbers by letting A be theset of all real numbers satisfying a x and letting B be the set of all real numbers b withx < b. Both A and B are nonempty because x is finite. Every a in A is below every bin B by transitivity of the order on the hyperreals. The completeness of the real numbersmeans that there is a real r at the gap between A and B. We must have x r, because ifx r > 1/m, say, then r + 1/(2m) < x and by the gap property would need to be in B.

A picture of the hyperreal number line looks like the ordinary real line at unit scale.We cant draw far enough to get to the infinitely large part and this theorem says eachfinite number is indistinguishably close to a real number. If we magnify or compress by newnumber amounts we can see new structure.

You still cannot divide by zero (that violates rules of algebra), but if is a positiveinfinitesimal, we can compute the following:

, 2, 1

What can we say about these quantities?

The idealization of infinitesimals lets us have our cake and eat it too. Since = 0, wecan divide by . However, since is tiny, 1/ must be HUGE.

Example 1.5. Negative infinitesimals

In ordinary algebra, if > 0, then < 0, so we can apply this rule to the infinitesimalnumber and conclude that < 0, since > 0.Example 1.6. Orders of infinitesimals

In ordinary algebra, if 0 < < 1, then 0 < 2 < , so 0 < 2 < .

We want you to formulate this more exactly in the next exercise. Just assume isvery small, but positive. Formulate what you want to draw algebraically. Try some smallordinary numbers as examples, like = 0.01. Plot at unit scale and place 2 accuratelyon the figure.

Example 1.7. Infinitely large numbers


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For real numbers if 0 < < 1/n then n < 1/. Since is infinitesimal, 0 < < 1/nfor every natural number n = 1, 2, 3, . . . Using ordinary rules of algebra, but substitutingthe infinitesimal , we see that H = 1/ > n is larger than any natural number n (or isinfinitely large), that is, 1 < 2 < 3 < .. . < n < H, for every natural number n. We cansee infinitely large numbers by turning the microscope around and looking in the otherend.

The new algebraic rules are the ones that tell us when quantities are infinitely close,a b. Such rules, of course, do not follow from rules about ordinary high school numbers,but the rules are intuitive and simple. More important, they let us calculate limits directly.

Theorem 1.12. Computation Rules for Finite and Infinitesimal Numbers(a) If p and q are finite, so are p + q and p q.(b) If and are infinitesimal, so is + .(c) If 0 and q is finite, then q 0. (finite x infsml = infsml)(d) 1/0 is still undefined and 1/x is infinitely large only when x 0.

To understand these rules, just think of p and q as fixed, if large, and as being assmall as you please (but not zero). It is not hard to give formal proofs from the definitionsabove, but this intuitive understanding is more important. The last rule can be seen onthe graph of y = 1/x. Look at the graph and move down near the values x 0.

x

y

Figure 1.2: y = 1/x

Proof:

We prove rule (c) and leave the others to the exercises. If q is finite, there is a naturalnumber m so that |q| < m. We want to show that |q | < 1/n for any natural number n.Since is infinitesimal, we have || < 1/(n m). By Exercise 1.2.5, |q| || < m 1nm = 1m .Example 1.8. y = x3 dy = 3x2 dx, for finite x

The error term in the increment of f[x] = x3, computed above is

= (x[3x + x])

Ifx is assumed finite, then 3x is also finite by the first rule above. Since 3x and x are finite,so is the sum 3x + x by that rule. The third rule, that says an infinitesimal times a finitenumber is infinitesimal, now gives x finite = x[3x + x] = infinitesimal, 0. This


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14 1. Numbers

justifies the local linearity ofx3 at finite values ofx, that is, we have used the approximationrules to show that

f[x + x] = f[x] + f[x] x + xwith 0 whenever x 0 and x is finite, where f[x] = x3 and f[x] = 3 x2.

Exercise set 1.4

1. Draw the view of the ideal number line when viewed under an infinitesimal microscopeof power 1/. Which number appears unit size? How big does 2 appear at this scale?Where do the numbers and 3 appear on a plot of magnification 1/2?

2. Backwards microscopes or compressionDraw the view of the new number line when viewed under an infinitesimal microscopewith its magnification reversed to power (not1/). What size does the infinitely largenumberH (HUGE) appear to be? What size does the finite (ordinary) number m = 109

appear to be? Can you draw the number H2 on the plot?

3. y = xp dy = p xp1 dx, p = 1, 2, 3, . . .For each f[x] = xp below:

(a) Compute f[x + x] f[x] and simplify, writing the increment equation:f[x + x] f[x] = f[x] x + x

= [term in x but not x]x + [observed microscopic error]x

Notice that we can solve the increment equation for =f[x + x] f[x]

x f[x]

(b) Show that 0 if x 0 and x is finite. Does x need to be finite, or can it beany hyperreal number and still have 0?

(1) If f[x] = x1, then f[x] = 1x0 = 1 and = 0.(2) If f[x] = x2, then f[x] = 2x and = x.(3) If f[x] = x3, then f[x] = 3x2 and = (3x + x)x.(4) If f[x] = x4, then f[x] = 4x3 and = (6x2 + 4xx + x2)x.(5) If f[x] = x5, then f[x] = 5x4 and = (10x3 + 10x2x + 5xx2 + x3)x.

4. Exceptional Numbers and the Derivative of y =1

x(a) Letf[x] = 1/x and show that

f[x + x] f[x]x

=1

x(x + x)

(b) Compute

=1

x(x + x)

+1

x2

= x

1

x2

(x + x)(c) Show that this gives

f[x + x] f[x] = f[x] x + xwhen f[x] = 1/x2.

(d) Show that 0 provided x is NOT infinitesimal (and in particular is not zero.)


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5. Exceptional Numbers and the Derivative of y =

x(a) Letf[x] =

x and compute

f[x + x] f[x] = 1x + x +

x

(b) Compute

=1

x + x +

x 1

2

x=

12

x(

x + x +

x)2 x

(c) Show that this gives

f[x + x] f[x] = f[x] x + xwhen f[x] = 1

2

x.

(d) Show that 0 provided x is positive and NOT infinitesimal (and in particularis not zero.)

6. Prove the remaining parts of Theorem 1.12.


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16 1. Numbers


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CHAPTER

2 Functional IdentitiesIn high school you learned that trig functions satisfy certain iden-tities or that logarithms have certain properties. This chapterextends the idea of functional identities from specific cases to adefining property of an unknown function.

The use of unknown functions is of fundamental importance in calculus, and otherbranches of mathematics and science. For example, differential equations can be viewed asidentities for unknown functions.

One reason that students sometimes have difficulty understanding the meaning of deriva-tives or even the general rules for finding derivatives is that those things involve equations inunknown functions. The symbolic rules for differentiation and the increment approximationdefining derivatives involve unknown functions. It is important for you to get used to thishigher type variable, an unknown function. This chapter can form a bridge between thespecific identities of high school and the unknown function variables from rules of calculusand differential equations.

2.1 Specific Functional IdentitiesAll the the identities you need to recall from high school are:

(Cos[x])2 + (Sin[x])2 = 1 CircleIden

Cos[x + y] = Cos[x]Cos[y] Sin[x]Sin[y] CosSumSin[x + y] = Sin[x]Cos[y] + Sin[y] Cos[x] SinSum

bx+y = bx by ExpSum

(bx

)y

= bxy

RepeatedExpLog[x y] = Log[x] + Log[y] LogProdLog[xp] = p Log[x] LogPower

but you must be able to use these identities. Some practice exercises using these familiaridentities are given in main text CD Chapter 28.

17


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18 2. Functional Identities

2.2 General Functional IdentitiesA general functional identity is an equation which is satisfied by an unknown

function (or a number of functions) over its domain.

The function

f[x] = 2x

satisfies f[x + y] = 2(x+y) = 2x2y = f[x]f[y], so eliminating the two middle terms, we seethat the function f[x] = 2x satisfies the functional identity

f[x + y] = f[x] f[y](ExpSum)

It is important to pay attention to the variable or variables in a functional identity. In order

for an equation involving a function to be a functional identity, the equation must be valid forall values of the variables in question. Equation (ExpSum) above is satisfied by the functionf[x] = 2x for all x and y. For the function f[x] = x, it is true that f[2 + 2] = f[2]f[2], butf[3 + 1] = f[3]f[1], so = x does not satisfy functional identity (ExpSum).

Functional identities are a sort of higher laws of algebra. Observe the notational simi-larity between the distributive law for multiplication over addition,

m (x + y) = m x + m yand the additive functional identity

f[x + y] = f[x] + f[y](Additive)

Most functions f[x] do not satisfy the additive identity. For example,

1

x + y= 1

x+

1

yand

x + y = x + y

The fact that these are not identities means that for some choices ofx and y in the domainsof the respective functions f[x] = 1/x and f[x] =

x, the two sides are not equal. You

will show below that the only differentiable functions that do satisfy the additive functionalidentity are the functions f[x] = m x. In other words, the additive functional identity isnearly equivalent to the distributive law; the only unknown (differentiable) function thatsatisfies it is multiplication. Other functional identities such as the 7 given at the start ofthis chapter capture the most important features of the functions that satisfy the respectiveidentities. For example, the pair of functions f[x] = 1/x and g[x] =

x do not satisfy the

addition formula for the sine function, either.

Example 2.1. The Microscope Equation

The microscope equation defining the differentiability of a function f[x] (see Chapter5 of the text),

f[x + x] = f[x] + f[x] x + x(Micro)


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General Functional Identities 19

with 0 if x 0, is similar to a functional identity in that it involves an unknownfunction f[x] and its related unknown derivative function f[x]. It relates the functionf[x] to its derivative df

dx= f

[x].

You should think of (Micro) as the definition of the derivative of f[x] at a given x, butalso keep in mind that (Micro) is the definition of the derivative of any function. If we letf[x] vary over a number of different functions, we get different derivatives. The equation(Micro) can be viewed as an equation in which the function, f[x], is the variable input, andthe output is the derivative dfdx .

To make this idea clearer, we rewrite (Micro) by solving for dfdx : )

df

dx=

f[x + x] f[x]x

or

df

dx= lim

x0f[x + x] f[x]

x

If we plug in the input function f[x] = x2 into this equation, the output is dfdx = 2x. If we

plug in the input function f[x] = Log[x], the output is dfdx =1x . The microscope equation

involves unknown functions, but strictly speaking, it is not a functional identity, becauseof the error term (or the limit which can be used to formalize the error). It is only anapproximate identity.

Example 2.2. Rules of Differentiation

The various differentiation rules, the Superposition Rule, the Product Rule and theChain Rule (from Chapter 6 of the text) are functional identities relating functions andtheir derivatives. For example, the Product Rule states:

d(f[x]g[x])

dx=

df

dxg[x] + f[x]

dg

dxWe can think of f[x] and g[x] as variables which vary by simply choosing different actualfunctions for f[x] and g[x]. Then the Product Rule yields an identity between the choicesof f[x] and g[x], and their derivatives. For example, choosing f[x] = x2 and g[x] = Log[x]and plugging into the Product Rule yields

d(x2 Log[x])

dx= 2x Log[x] + x2

1

x

Choosing f[x] = x3 and g[x] = Exp[x] and plugging into the Product Rule yields

d(x3 Exp[x])

dx= 3x2 Exp[x] + x3 Exp[x]

If we choose f[x] = x5, but do not make a specific choice for g[x], plugging into theProduct Rule will yield

d(x5g[x])

dx= 5x4g[x] + x5

dg

dx

The goal of this chapter is to extend your thinking to identities in unknown functions.


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Exercise set 2.2

1. (a) Verify that for any positive number, b, the function f[x] = bx satisfies the func-

tional identity (ExpSum) above for all x and y.(b) Is (ExpSum) valid (for all x and y) for the function f[x] = x2 or f[x] = x3?

Justify your answer.

2. Define f[x] = Log[x] where x is any positive number. Why does this f[x] satisfy thefunctional identities

f[x y] = f[x] + f[y](LogProd)and

f[xk] = kf[x](LogPower)

where x, y, and k are variables. What restrictions should be placed on x and y for theabove equations to be valid? What is the domain of the logarithm?

3. Find values of x and y so that the left and right sides of each of the additive formulasfor 1/x and

x above are not equal.

4. Show that 1/x and

x also do not satisfy the identity (SinSum), that is,

1

x + y=

1

x

y +

x

1

y

is false for some choices of x and y in the domains of these functions.

5. (a) Suppose that f[x] is an unknown function which is known to satisfy (LogProd)(so f[x] behaves like Log[x], but we dont know if f[x] is Log[x]), and supposethat f[0] is a well-defined number (even though we dont specify exactly what f[0]is). Show that this function f[x] must be the zero function, that is show thatf[x] = 0 for every x. (Hint: Use the fact that 0 x = 0).

(b) Suppose that f[x] is an unknown function which is known to satisfy (LogPower)for allx > 0 and all k. Show that f[1] must equal 0, f[1] = 0. (Hint: Fix x = 1,and try different values of k).

6. (a) Letm and b be fixed numbers and define

f[x] = m x + b

Verify that if b = 0, the above function satisfies the functional identity

f[x] = x f[1](Mult)

for allx and that if b = 0, f[x] will not satisfy (Mult) for all x (that is, given anonzero b, there will be at least one x for which (Mult) is not true).

(b) Prove that any function satisfying (Mult) also automatically satisfies the twofunctional identities


and

f[x y] = x f[y](Multiplicative)

for allx and y.


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The Function Extension Axiom 21

(c) Suppose f[x] is a function which satisfies (Mult) (and for now that is the onlything you know about f[x]). Prove that f[x] must be of the form f[x] = m x,

for some fixed numberm (this is almost obvious).(d) Prove that a general power function, f[x] = mxk where k is a positve integer and

m is a fixed number, will not satisfy (Mult) for all x if k = 1, (that is, if k = 1,there will be at least one x for which (Mult) is not true).

(e) Prove that f[x] = Sin[x] does not satisfy the additive identity.(f) Prove that f[x] = 2x does not satisfy the additive identity.

7. (a) Let f[x] and g[x] be unknown functions which are known to satisfy f[1] = 2,dfdx (1) = 3, g(1) = 3, dgdx (1) = 4. Let h(x) = f[x]g[x]. Compute dhdx (1).

(b) Differentiate the general Product Rule identity to get a formula for

d2(f g)

dx2

Use your rule to compute d2(h)

dx2 (1) if

d2(f)

dx2 (1) = 5 and

d2(g)

dx2 (1) =

2, using other

values from part 1 of this exercise.

2.3 The Function Extension AxiomThis section shows that all real functions have hyperreal extensions that arenatural from the point of view of properties of the original function.

Roughly speaking, the Function Extension Axiom for hyperreal numbers says that the

natural extension of any real function obeys the same functional identities and inequalitiesas the original function. In Example 2.7, we use the identity,

f[x + x] = f[x] f[x]with x hyperreal and x 0 infinitesimal where f[x] is a real function satisfying f[x + y] =f[x] f[y]. The reason this statement of the Function Extension Axiom is rough is becausewe need to know precisely which values of the variables are permitted. Logically, we canexpress the axiom in a way that covers all cases at one time, but this is a little complicatedso we will precede that statement with some important examples.

The Function Extension Axiom is stated so that we can apply it to the Log identity inthe form of the implication

(x > 0 & y > 0)

Log[x] and Log[y] are defined and Log[x

y] = Log[x] + Log[y]

The natural extension of Log[] is defined for all positive hyperreals and its identities hold forhyperreal numbers satisfying x > 0 and y > 0. The other identities hold for all hyperreal xand y. To make all such statements implications, we can state the exponential sum equationas

(x = x & y = y) ex+y = ex ey


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The differential

d(Sin[]) = Cos[] d

is a notational summary of the valid approximation

Sin[ + ] Sin[] = Cos[] + where 0 when 0. The derivation of this approximation based on magnifying acircle (given in a CD Section of Chapter 5 of the text) can be made precise by using theFunction Extension Axiom in the place where it locates (Cos[ + ], Sin[ + ]) on the unitcircle. This is simply using the extension of the (CircleIden) identity to hyperreal numbers,(Cos[ + ])2 + (Sin[ + ])2 = 1.Logical Real Expressions, Formulas and Statements

Logical real expressions are built up from numbers and variables using functions. Hereis the precise definition.

(a) A real number is a real expression.

(b) A variable standing alone is a real expression.(c) If E1, E2, , En are a real expressions and f[x1, x2, , xn] is a real function ofn

variables, then f[E1, E2, , En] is a real expression.A logical real formula is one of the following:

(a) An equation between real expressions, E1 = E2.(b) An inequality between real expressions, E1 < E2, E1 E2, E1 > E2, E1 E2, or

E1 = E2.(c) A statement of the form E is defined or of the form E is undefined.

Let S and T be finite sets of real formulas. A logical real statement is an implication of theform,

S T

or whenever every formula in S is true, then every formula in T is true.Logical real statements allow us to formalize statements like: Every point in the square

below lies in the circle below. Formalizing the statement does not make it true or false.Consider the figure below.

x

y

Figure 2.1: Square and Circle


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The Function Extension Axiom 23

The inside of the square shown can be described formally as the set of points satisfying theequations in the set S = { 0 x, 0 y, x 1.2, y 1.2 }. The inside of the circle shown canbe defined as the set of points satisfying the single equation T =

{(x

1)2 +(y

1)2

1.62

}.

This is the circle of radius 1.6 centered at the point (1, 1). The logical real statement S Tmeans that every point inside the square lies inside the circle. The statement is true forevery real x and y. First of all, it is clear by visual inspection. Second, points (x, y) thatmake one or more of the formulas in S false produce a false premise, so no matter whetheror not they lie in the circle, the implication is logically true (if uninteresting).

The logical real statement T S is a valid logical statement, but it is false since itsays every point inside the circle lies inside the square. Naturally, only true logical realstatements transfer to the hyperreal numbers.

Axiom 2.1. The Function Extension AxiomEvery logical real statement that holds for all real numbers also holds for all hyper-real numbers when the real functions in the statement are replaced by their naturalextensions.

The Function Extension Axiom establishes the 5 identities for all hyperreal numbers,because x = x and y = y always holds. Here is an example.

Example 2.3. The Extended Addition Formula for Sine

S = {x = x, y = y} T = { Sin[x] is defined ,Sin[y] is defined ,

Cos[x] is defined ,

Cos[y] is defined ,

Sin[x + y] = Sin[x] Cos[y] + Sin[y] Cos[x]}The informal interpretation of the extended identity is that the addition formula for sineholds for all hyperreals.

Example 2.4. The Extended Formulas for Log

We may take S to be formulas x > 0, y > 0 and p = p and T to be the functionalidentities for the natural log plus the statements Log[ ] is defined, etc. The FunctionExtension Axiom establishes that log is defined for positive hyperreals and satisfies the twobasic log identities for positive hyperreals.

Example 2.5. Abstract Uses of Function Extension

There are two general uses of the Function Extension Axiom that underlie most of thetheoretical problems in calculus. These involve extension of the discrete maximum and

extension of finite summation. The proof of the Extreme Value Theorem 4.4 below uses ahyperfinite maximum, while the proof of the Fundamental Theorem of Integral Calculus 5.1uses hyperfinite summation.

Equivalence of infinitesimal conditions for derivatives or limits and the epsilon - deltareal number conditions are usually proved by using an auxiliary real function as in the proofof the limit equivalence Theorem 3.2.


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Example 2.6. The Increment Approximation

Note: The increment approximation

f[x + x] = f[x] + f[x] x + x

with 0 for x 0 and the simpler statement

x 0 f[x] f[x] + x) f[x]x

are not real logical expressions, because they contain the relation , which is not includedin the formation rules for logical real statements. (The relation does not apply to ordinaryreal numbers, except in the trivial case x = y.)

For example, if is any hyperreal and 0, thenSin[ + ] = Sin[] Cos[] + Sin[] Cos[]

by the natural extension of the addition formula for sine above. Notice that the naturalextension does NOT tell us the interesting and important estimate

Sin[ + ] = Sin[] + Cos[] +

with 0 when 0. (I.e., Cos[] = 1 + and Sin[]/ 1 are true, but not reallogical statements we can derive just from natural extensions.)

Exercise set 2.3

1. Write a formal logical real statement S

T that says, Every point inside the circle

of radius 2, centered at (1, 3) lies outside the square with sides x = 0, y = 0, x = 1,y = 1. Draw a figure and decide whether or not this is a true statement for all realvalues of the variables.

2. Write a formal logical real statement S T that is equivalent to each of the functionalidentities on the first page of the chapter and interpret the extended identities in thehyperreals.

2.4 Additive FunctionsAn identity for an unknown function together with the increment approxi-mation combine to give a specific kind of function. The two ideas combineto give a differential equation. After you have learned about the calculusof the natural exponential function in Chapter 8 of the text, you will easilyunderstand the exact solution of the problem of this section.


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Additive Functions 25

In the early 1800s, Cauchy asked the question: Must a function satisfying


be of the form f[x] = m x? This was not solved until the late 1800s by Hamel. The answeris No. There are some very strange functions satisfying the additive identity that are notsimple linear functions. However, these strange functions are not differentiable. We willsolve a variant of Cauchys problem for differentiable functions.

Example 2.7. A Variation on Cauchys Problem

Suppose an unknown differentiable function f[x] satisfies the (ExpSum) identity for allx and y,

f[x + y] = f[x] f[y]Does the function have to be f[x] = bx for some positive b?

Since our unknown function f[x] satisfies the (ExpSum) identity and is differentiable,

both of the following equations must hold:f[x + y] = f[x] f[y]f[x + x] = f[x] + f[x] x + x

We let y = x in the first identity to compare it with the increment approximation,

f[x + x] = f[x] f[x]f[x + x] = f[x] + f[x] x + x

so

f[x] f[x] = f[x] + f[x] x + xf[x][f[x] 1] = f[x] x + x

f[x] = f[x] f[x] 1x

or

f[x]f[x]

=f[x] 1

x

with 0 when x 0. The identity still holds with hyperreal inputs by the FunctionExtension Axiom. Since the left side of the last equation depends only on x and the right

hand side does not depend on x at all, we must have f[x]1x k, a constant, or f[x]1x kas x 0. In other words, a differentiable function that satisfies the (ExpSum) identitysatisfies the differential equation

df

dx= k f

What is the value of our unknown function at zero, f[0]? For any x and y = 0, we have

f[x] = f[x + 0] = f[x] f[0]so unless f[x] = 0 for all x, we must have f[0] = 1.

One of the main morals of this course is that if you know:(1) where a quantity starts,


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and(2) how a quantity changes,

then you can compute subsequent values of the quantity. In this problem we have found(1) f[0] = 1 and (2) dfdx = k f. We can use this information with the computer to calculatevalues of our unknown function f[x]. The unique symbolic solution to

f[0] = 1

df

dx= k f

is

f[x] = ek x

The identity (Repeated Exp) allows us to write this as

f[x] = ek x = (ek)x = bx

where b = ek. In other words, we have shown that the only differentiable functions thatsatisfy the (ExpSum) identity are the ones you know from high school, bx.

Problem 2.1. Smooth Additive Functions ARE LinearH

Suppose an unknown function is additive and differentiable, so it satisfies both

f[x + x] = f[x] + f[x](Additive)

and

f[x + x] = f[x] + f[x] x + x(Micro)Solve these two equations for f[x] and argue that since the right side of the equation does

not depend on x, f[x] must be constant. (Or f[x]x f[x1] and f[x]x f[x2], but sincethe left hand side is the same, f[x1] = f[x2].)

What is the value of f[0] if f[x] satisfies the additive identity?The derivative of an unknown function f[x] is constant and f[0] = 0, what can we say

about the function? (Hint: Sketch its graph.)N

A project explores this symbolic kind of linearity and the microscope equation fromanother angle.

2.5 The Motion of a PendulumDifferential equations are the most common functional identities which arisein applications of mathematics to solving real world problems. One of thevery important points in this regard is that you can often obtain significantinformation about a function if you know a differential equation the functionsatisfies, even if you do not know an exact formula for the function.


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The Motion of a Pendulum 27

For example, suppose you know a function [t] satisfies the differential equation

d2

dt2 = Sin[[t]]This equation arises in the study of the motion of a pendulum and [t] does not have aclosed form expression. (There is no formula for [t].) Suppose you know [0] = 2 . Thenthe differential equation forces

d2

dt2[0] = Sin[[0]] = Sin[

2] = 1

We can also use the differential equation for to get information about the higher deriva-tives of [t]. Say we know that ddt [0] = 2. Differentiating both sides of the differentialequation yields

d3

dt3= Cos[[t]]

d

dtby the Chain Rule. Using the above information, we conclude that

d3

dt3[0] = Cos[[0]]

d

dt[0] = Cos[

2]2 = 0

Problem 2.2.H

Derive a formula for d4

dt4 and prove thatd4dt4 [0] = 1.

N


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-4 -2 2 4w

-4

-2

2

4

x

Part 2

Limits


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30


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CHAPTER

3 The Theory of LimitsThe intuitive notion of limit is that a quantity gets close to a lim-iting value as another quantity approaches a value. This chapterdefines two important kinds of limits both with real numbers andwith hyperreal numbers. The chapter also gives many computationsof limits.

A basic fact about the sine function is

limx0

Sin[x]

x= 1

Notice that the limiting expression Sin[x]x is defined for 0 < |x0| < 1, but not ifx = 0. Thesine limit above is a difficult and interesting one. The important topic of this chapter is,What does the limit expression mean? Rather than the more practical question, Howdo I compute a limit?

Here is a simpler limit where we can see what is being approached.

limx1

x2 1x 1 = 2

While this limit expression is also only defined for 0 < |x 1|, or x = 1, the mystery iseasily resolved with a little algebra,

x2 1x 1 =

(x 1)(x + 1)(x 1) = x + 1

So,

limx1x2

1

x 1 = limx1(x + 1) = 2The limit limx1(x + 1) = 2 is so obvious as to not need a technical definition. If x is

nearly 1, then x + 1 is nearly 1 + 1 = 2. So, while this simple example illustrates that theoriginal expression does get closer and closer to 2 as x gets closer and closer to 1, it skirtsthe issue of how close?

31


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32 3. The Theory of Limits

3.1 Plain LimitsTechnically, there are two equivalent ways to define the simple continuousvariable limit as follows.

Definition 3.1. LimitLetf[x] be a real valued function defined for0 < |xa| < with a fixed positivereal number. We say

limxa

f[x] = b

when either of the equivalent the conditions of Theorem 3.2 hold.

Theorem 3.2. Limit of a Real VariableLetf[x] be a real valued function defined for0 < |xa| < with a fixed positivereal number. Let b be a real number. Then the following are equivalent:

(a) Whenever the hyperreal number x satisfies 0 < |x a| 0, the naturalextension function satisfies

f[x] b(b) For every accuracy tolerance there is a sufficiently small positive real num-

ber such that if the real number x satisfies 0 < |x a| < , then|f[x] b| <

Proof:

We show that (a) (b) by proving that not (b) implies not (a), the contrapositive.Assume (b) fails. Then there is a real > 0 such that for every real > 0 there is a real xsatisfying 0 < |x a| < and |f[x] b| . Let X[] = x be a real function that choosessuch an x for a particular . Then we have the equivalence

{ > 0} {X[] is defined , 0 < |X[] a| < , |f[X[] b| }

By the Function Extension Axiom 2.1 this equivalence holds for hyperreal numbers andthe natural extensions of the real functions X[] and f[]. In particular, choose a positiveinfinitesimal and apply the equivalence. We have 0 < |X[] a| < and |f[X[] b| > and is a positive real number. Hence, f[X[]] is not infinitely close to b, proving not (a)and completing the proof that (a) implies (b).

Conversely, suppose that (b) holds. Then for every positive real , there is a positive real

such that 0 < |x a| < implies |f[x] b| < . By the Function Extension Axiom 2.1,this implication holds for hyperreal numbers. If a, then 0 < | a| < for every real ,so |f[] b| < for every real positive . In other words, f[] b, showing that (b) implies(a) and completing the proof of the theorem.

Example 3.1. Condition (a) Helps Prove a Limit


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Plain Limits 33

Suppose we wish to prove completely rigorously that

limx01

2(2 + x) =

1

4

The intuitive limit computation of just setting x = 0 is one way to see the answer,

limx0

1

2(2 + x)=

1

2(2 + 0=

1

4

but this certainly does not demonstrate the epsilon - delta condition (b).Condition (a) is almost as easy to establish as the intuitive limit computation. We wish

to show that when x 01

2(2 + x) 1

4

Subtract and do some algebra,

12(2 + x)

14

= 24(2 + x)

(2 + x)4(2 + x)

=x

4(2 + x)= x 1

4(2 + x)

We complete the proof using the computation rules of Theorem 1.12. The fraction 1/(4(2+x)) is finite because 4(2 + x) 8 is not infinitesimal. The infinitesimal x times a finitenumber is infinitesimal.

1

2(2 + x) 1

4 0

1

2(2 + x) 1

4

This is a complete rigorous proof of the limit. Theorem 3.2 shows that the epsilon - deltacondition (b) holds.

Exercise set 3.1

1. Prove rigorously that the limit limx0 13(3+x) =19 . Use your choice of condition (a)

or condition (b) from Theorem 3.2.

2. Prove rigorously that the limit limx0 14+x+

4= 14 . Use your choice of condition

(a) or condition (b) from Theorem 3.2.

3. The limit limx0Sin[x]

x = 1 means that sine of a small value is nearly equal to the value,

and near in a strong sense. Suppose the natural extension of a functionf[x] satisfiesf[] 0 whenever 0. Does this mean that limx0 f[x]x exists? (HINT: What islimx0

x? What is

/?)

4. Assume that the derivative of sine is cosine and use the increment approximation

f[x + x] f[x] = f[x] x + x


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34 3. The Theory of Limits

with 0 when x 0, to prove the limit limx0 Sin[x]x = 1. (It means essentially thesame thing as the derivative of sine at zero is 1. HINT: Take x = 0 and x = x in theincrement approximation.)

3.2 Function LimitsMany limits in calculus are limits of functions. For example, the derivativeis a limit and the derivative of x3 is the limit function 3 x2. This sectiondefines the function limits used in differentiation theory.

Example 3.2. A Function Limit

The derivative ofx3 is 3 x2, a function. When we compute the derivative we use the limit

limx0

(x + x)3 x3x

Again, the limiting expression is undefined at x = 0. Algebra makes the limit intuitivelyclear,

(x + x)3 x3x

=(x3 + 3 x2 x + 3 x x2 + x3) x3

x= 3 x2 + 3 x x + x2

The terms with x tend to zero as x tends to zero.

limx0

(x + x)3 x3x

= limx0

(3 x2 + 3 x x + x2) = 3 x2

This is clear without a lot of elaborate estimation, but there is an important point thatmight be missed if you dont remember that you are taking the limit of a function. Thegraph of the approximating function approaches the graph of the derivative function. Thismore powerful approximation (than that just a particular value of x) makes much of thetheory of calculus clearer and more intuitive than a fixed x approach. Intuitively, it is noharder than the fixed x approach and infinitesimals give us a way to establish the uniformtolerances with computations almost like the intuitive approach.

Definition 3.3. Locally Uniform Function LimitLetf[x] and F[x, x] be real valued functions defined when x is in a real interval(a, b) and 0 < x < with a fixed positive real number. We say

limx0

F[x, x] = f[x]

uniformly on compact subintervals of (a, b), or locally uniformly when one of theequivalent the conditions of Theorem 3.4 holds.


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Function Limits 35

Theorem 3.4. Limit of a Real FunctionLetf[x] and F[x, x] be real valued functions defined when x is in a real interval

(a, b) and 0 < x < with a fixed positive real number. Then the following areequivalent:(a) Whenever the hyperreal numbers x and x satisfy 0 < |x| 0, x is finite,

and a < x < b with neither x a nor x b, the natural extension functionssatisfy

F[x,x] f[x](b) For every accuracy tolerance and every real and in (a, b), there is

a sufficiently small positive real number such that if the real number xsatisfies 0 < |x| < and the real number x satisfies x , then

|F[x, x] f[x]| <

Proof:

First, we prove not (b) implies not (a). If (b) fails, there are real and , a < < < b,and real positive such that for every real positive there are x and x satisfying

0 < x < , x , |F[x, x] f[x]|

Define real functions X[] and DX[] that select such values of x and x,

0 < DX[] < , X[] , |F[X[], DX[]] f[X[]]|

Now apply the Function Extension Axiom 2.1 to the equivalent sets of inequalities,

{ > 0} {0 < DX[] < , X[] , |F[X[], DX[]] f[X[]]| }Choose an infinitesimal 0 and let x = X[] and x = DX[]. Then

0 < x < 0, x , |F[x,x] f[x]|

so F[x,x] f[x] is not infinitesimal showing not (a) holds and proving (a) implies (b).Now we prove that (b) implies (a). Let x be a non zero infinitesimal and let x satisfy

the conditions of (a). We show that F[x,x] f[x] by showing that for any

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